Q.Charcoal pieces of a tree found at an archaeological site have a carbon-14 content that is only 17.5% that of an equivalent sample of carbon from a living tree. What is the age of the tree?

Radiocarbon (carbon) dating is a practical application of beta-decay physics used to estimate the age of ancient organic remains. Every living organism continuously absorbs carbon dioxide from the atmosphere to build its own organic molecules; while the great majority of this absorbed carbon is the stable isotope $^{12}_{6}C$, a very small, nearly constant fraction ($1.3\times10^{-12}$) is the radioactive isotope $^{14}_{6}C$, whose half-life is 5730 years. Atmospheric $^{14}_{6}C$ is continuously decaying away, but it is simultaneously being continuously replenished by cosmic rays striking the upper atmosphere, so the ratio of $^{14}_{6}C$ to $^{12}_{6}C$ stays essentially constant in the atmosphere - and, through continuous absorption, in every living organism - for as long as that organism remains alive.

Once an organism dies, it stops absorbing new carbon dioxide, so its existing $^{14}_{6}C$ is no longer topped up and simply decays away following the ordinary exponential decay law; the ratio of $^{14}_{6}C$ to $^{12}_{6}C$ in the dead remains therefore steadily falls over subsequent centuries and millennia. Measuring a sample's present-day carbon-14 activity $R$, and knowing (or estimating from atmospheric abundance) what its activity $R_0$ would have been when it was still part of a living organism, the elapsed time since death - the sample's age - follows directly from the activity form of the decay law, $R=R_0e^{-\lambda t}$, rearranged to …